Of course he does.
The point is that Batman is the archetype of a right-wing superhero. Batman is how rightwingers understand social justice: accumulate as much wealth as you can, use crushing physical violence to punish bad guys, act charitably at an individual level but do not ever work to solve social issues at a systemic level.
Even in-universe he's nowhere near as much of a positive force as he could be if he used his money to force political and social change instead of as an outlet for his mental issues.
He's not actively villainous because right-wingers don't see themselves as such. But when that fantasy meets reality, you get Elon Musk.